You have found the following ages (in years) of 6 snakes. The snakes are randomly selected from the 45 snakes at your local zoo: $ 31,\enspace 22,\enspace 34,\enspace 10,\enspace 2,\enspace 25$ Based on your sample, what is the average age of the snakes? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 45 snakes, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{106.09} + {1.69} + {176.89} + {114.49} + {349.69} + {18.49}} {{6 - 1}} $ $ {s^2} = \dfrac{{767.34}}{{5}} = {153.47\text{ years}^2} $ We can estimate that the average snake at the zoo is 20.7 years old. There is a variance of 153.47 years $^2$.